
The negation of a proposition \(p\)  is defined by the authors as follows: “since all operators of quantum mechanics are Hermitian, the mathematical representative of the negative of any experimental proposition is the orthogonal complement of the mathematical representative of the proposition itself” ([1]}, pp. 826-827). The orthogonal complement \(P^{\perp }\)  of the subspace \(P\)  is the set whose elements are all the vectors orthogonal to the elements of \(P\) . Such an orthogonal complement satisfies the following property: given a subset \(P\subset \mathcal {H}\)  and a pure state \(\psi \) , \(\psi (P)=1\)  iff \(\psi (P^{\perp })=0\)  and \(\psi (P)=0\)  iff \(\psi (P^{\perp })=1\) . As Dalla Chiara and Giuntini underline, “\(\psi \)  assigns to an event probability 1 (0, respectively) iff \(\psi \)  assigns to the orthocomplement of \(P\)  [notation adapted] probability 0 (1, respectively). As a consequence, one is dealing with an operation that inverts the two extreme probability-values, which naturally correspond to the truth-values truth and falsity (similarly to the classical truth-table of negation)” ([2]}, p. 132).

Concerning the conjunction, Birkhoff and von Neumann note that one can retain the very same set-theoretical interpretation of the classical conjunction in QL, since the intersection of two closed subspaces \(P, Q\subset \mathcal {H}\)  is still a closed subspace. Thus, one maintains the usual meaning for the conjunction: the pure state \(\psi \)  verifies \(P\cap Q\)  iff \(\psi \)  verifies both \(P\)  and \(Q\) . Thus, quantum logic does not introduce a new logical operator for the conjunction.

Contrary to the previous case, the logical operator for disjunction cannot be represented by the set-theoretic union, since the set-theoretic union of the two subspaces \(P, Q\)  will not be in general a subspace; thus, it is not an experimental proposition. Therefore, in QL one introduces the quantum logical disjunction as the the closed span of the subspaces \(P, Q\) , which is an experimental proposition. Such a statement corresponds to the “smallest closed subspace containing both \(P\)  and \(Q\) ” ([3]}, p. 55).

As far as logical implication is concerned, [1]} claim that given two experimental propositions \(p\)  and \(q\)  about a certain physical system, “\(p\)  implies \(q\) ” means that “whenever one can predict \(p\)  with certainty, one can predict also \(q\)  with certainty” (p. 827) and this is equivalent to stating that “the mathematical representative of \(p\)  is a subset of the mathematical representative of \(q\) ” (ibid.). This fact is particularly important since the authors showed that “it is algebraically reasonable to try to correlate physical qualities with subsets of phase-space” and thus, “physical qualities attributable to any physical system form a partially ordered system” ([1]}, p. 828, notation adapted).

